NODES format

If the problem dimensions may depend on past events, then a pathwise description of the problem becomes cumbersome. SMPS allows a node by node construction of the event tree, as shown in this example. A multi-period production and inventory problem is set up, where the number of products depends on previous events. For instance, if the demand for a particular product exceeds a threshold, then a complementary product may be introduced; conversely, if the demand is low, a product may be discontinued. A problem of this type was introduced in Gassmann and Schweitzer[1].

The algebraic formulation is as follows:

where H^t–1 is the history vector, i.e., H^t–1=(s₀,…,s_t–1),

S_t(H^t–1) is the set of branches in the event tree contingent on observing the history H^t–1,

p_t(H^t–1,s_t) is the probability of observing event sequence (H^t–1,s_t),

I_t(H^t–1,s_t) is the set of products produced during period t under the event sequence (H^t–1,s_t),

J_t(H^t–1,s_t) is the set of resources needed for the scheduled production during period t under the event sequence (H^t–1,s_t),

h_it is the cost of dealing with one unsold unit of product i during period t, either holding it in inventory or disposing of it if production ceases.

r_it(H^t–1,s_t) is the revenue of selling one unit of product i during period t under the event sequence (H^t–1,s_t),

c_it(H^t–1,s_t) is the cost of producing one unit of product i during period t under the event sequence (H^t–1,s_t),

g_it(H^t–1,s_t) is the cost of being short one unit of product i during period t under the event sequence (H^t–1,s_t),

d_it(H^t–1,s_t) is the observed demand for product i under the event sequence (H^t–1,s_t),

v_i is the storage capacity required to store one unit of product i,

C is the total available storage capacity,

a_ij is the amount of resource j required to produce one unit of product i,

M_jt(H^t–1,s_t) is the amount of resource j available in period t under the event sequence (H^t–1,s_t),

p_it(H^t–1,s_t) is the amount of product i produced in period t under the event sequence (H^t–1,s_t),

x_it(H^t–1,s_t) is the amount of product i left unsold in period t under the event sequence (H^t–1,s_t),

y_it(H^t–1,s_t) is the amount of discontinued product i remaining in period t under the event sequence (H^t–1,s_t),

z_it(H^t–1,s_t) is the amount of product i sold in period t under the event sequence (H^t–1,s_t),

d_it(H^t–1,s_t) = 0 if product i was newly introduced in period t–1 under the event sequence (H^t–1,s_t),

= 1 if product i was also produced in the previous period.

Time file

Core file

Stoch file

MPL file

Reference

[1] H.I. Gassmann and E. Schweitzer, “A comprehensive input format for stochastic linear programs”, Annals of Operations Research 104 (2001) 89–125.